Characterization of $k$-positive maps

We present a general characterization of k-positivity for a positive map in terms of the estimation of the Ky Fan norm of the matrix constructed from the Kraus operators of the associated completely positive map. Combining this with the result given by Takasaki and Tomiyama we construct a family of positive maps between matrix algebras of different dimensions depending on a parameter. The estimate bounds on the parameter to obtain the $k$-positivity are better than those derived from the spectral conditions considered by Chru\'sci\'nski and Kossakowski. We further look with special attention at the case where we give the precise bound for the regions of decomposability.